The algorithmic numbers in non-archimedean numerical computing environments

Abstract

<p style='text-indent:20px;'>There are many natural phenomena that can best be described by the use of infinitesimal and infinite numbers (see e.g. [<xref ref-type="bibr" rid="b1">1</xref>,<xref ref-type="bibr" rid="b5">5</xref>,<xref ref-type="bibr" rid="b13">13</xref>,<xref ref-type="bibr" rid="b23">23</xref>]. However, until now, the Non-standard techniques have been applied to theoretical models. In this paper we investigate the possibility to implement such models in numerical simulations. First we define the field of Euclidean numbers which is a particular field of hyperreal numbers. Then, we introduce a set of families of Euclidean numbers, that we have called altogether <i>algorithmic numbers</i>, some of which are inspired by the IEEE 754 standard for floating point numbers. In particular, we suggest three formats which are relevant from the hardware implementation point of view: the Polynomial Algorithmic Numbers, the Bounded Algorithmic Numbers and the Truncated Algorithmic Numbers. In the second part of the paper, we show a few applications of such numbers.